The complexity of quantum spin systems on a two-dimensional square lattice

Abstract

The problem 2-LOCAL HAMILTONIAN has been shown to be complete for the quantum computational class QMA, see quant-ph/0406180. In this paper we show that this important problem remains QMA-complete when the interactions of the 2-local Hamiltonian are between qubits on a two-dimensional (2-D) square lattice. Our results are partially derived with novel perturbation gadgets that employ mediator qubits which allow us to manipulate k-local interactions. As a side result, we obtain that quantum adiabatic computation using 2-local interactions restricted to a 2-D square lattice is equivalent to the circuit model of quantum computation. Our perturbation method also shows how any stabilizer space associated with a k-local stabilizer (for constant k) can be generated as an approximate ground-space of a 2-local Hamiltonian.

19 pages, 12 figures; v2 has 23 pages, 12 figures and some additional results in an appendix. The presentation in v3 is changed considerably. v4 has various corrections and an improved presentation. v5 has an additional correction. v6 is close to final, sigh. v6 uses qic.sty and is to appear in QIC

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